8,268 research outputs found
Proper Actions of Automorphism Groups of Free Products of Finite Groups
If G is a free product of finite groups, let ΣAut1(G) denote all (necessarily symmetric) automorphisms of G that do not permute factors in the free product. We show that a McCullough–Miller and Gutiérrez–Krstić derived (also see Bogley–Krstić) space of pointed trees is an EΣAut1(G)-space for these groups
On the structure of quantum automorphism groups
We compute the K-theory of quantum automorphism groups of finite dimensional C∗-algebras in the sense of Wang. The results show in particular that the C∗-algebras of functions on the quantum permutation groups S+n are pairwise non-isomorphic for different values of n. Along the way we discuss some general facts regarding torsion in discrete quantum groups. In fact, the duals of quantum automorphism groups are the most basic examples of discrete quantum groups exhibiting genuine quantum torsion phenomena
Dimension invariants of outer automorphism groups
The geometric dimension for proper actions of a
group is the minimal dimension of a classifying space for proper actions
. We construct for every integer , an example of a
virtually torsion-free Gromov-hyperbolic group such that for every group
which contains as a finite index normal subgroup, the virtual
cohomological dimension of equals
but such that the outer automorphism group
is virtually torsion-free, admits a cocompact model for
but nonetheless has
.Comment: 24 page
Automorphisms of two-dimensional right-angled Artin groups
We study the outer automorphism group of a right-angled Artin group A_G in
the case where the defining graph G is connected and triangle-free. We give an
algebraic description of Out(A_G) in terms of maximal join subgraphs in G and
prove that the Tits' alternative holds for Out(A_G). We construct an analogue
of outer space for Out(A_G) and prove that it is finite dimensional,
contractible, and has a proper action of Out(A_G). We show that Out(A_G) has
finite virtual cohomological dimension, give upper and lower bounds on this
dimension and construct a spine for outer space realizing the most general
upper bound.Comment: Bounds on vcd improved, proof of Tits' alternative added, expository
improvements, typos correcte
Finite and infinite quotients of discrete and indiscrete groups
These notes are devoted to lattices in products of trees and related topics.
They provide an introduction to the construction, by M. Burger and S. Mozes, of
examples of such lattices that are simple as abstract groups. Two features of
that construction are emphasized: the relevance of non-discrete locally compact
groups, and the two-step strategy in the proof of simplicity, addressing
separately, and with completely different methods, the existence of finite and
infinite quotients. A brief history of the quest for finitely generated and
finitely presented infinite simple groups is also sketched. A comparison with
Margulis' proof of Kneser's simplicity conjecture is discussed, and the
relevance of the Classification of the Finite Simple Groups is pointed out. A
final chapter is devoted to finite and infinite quotients of hyperbolic groups
and their relation to the asymptotic properties of the finite simple groups.
Numerous open problems are discussed along the way.Comment: Revised according to referee's report; definition of BMW-groups
updated; more examples added in Section 4; new Proposition 5.1
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